Double Lenses

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Abstract
The analysis of the shear induced by a single cluster, acting as a gravitational lens, on the images of a large number of background galaxies is centered around the curl-free character of a well-known vector field that can be derived from the data; in particular, the mass reconstruction methods, currently producing many interesting astrophysical applications, are all based on this tensorial property of the induced shear. Such a basic property breaks down when the source galaxies happen to be observed through two clusters at different redshifts, partially aligned along the line of sight, because an asymmetric part in the Jacobian matrix associated with the ray-tracing transformation is expected, now that the incoming light rays are bent twice. In this paper we address the study of double lenses and obtain five main results. (1) First we generalize the procedure to extract the available information, contained in the observed shear field, from the case of a single lens to that of a double lens. (2) Then we evaluate the possibility of detecting the signature of double lensing given the known properties of the distribution of clusters of galaxies. In particular, we show that a few configurations are likely to be present in the sky, for which the small effects characteristic of double lensing may already be within detection limits (i.e., we show that the signal involved is expected to be larger than what could be produced by statistical noise, which includes effects associated with the distribution of the source ellipticities and with the spread in redshift of the lensed galaxies). (3) As a different astrophysical application, we demonstrate how the method can be used to detect the presence of a dark cluster that might happen to be partially aligned with a bright cluster studied in terms of statistical lensing; if the properties of the bright cluster are well constrained by independent diagnostics, the location and the structure of the dark cluster can be reconstructed. (4) In addition, we show that the redshift distribution of the source galaxies, which in principle might also contribute to break the curl-free character of the shear field, actually produces systematic effects typically 2 orders of magnitude smaller than the double-lensing effects we are focusing on. (5) Remarkably, a discussion of relevant contributions to the noise of the shear measurement has brought up an intrinsic limitation of weak-lensing analyses, since one specific contribution, associated with the presence of a nonvanishing two-galaxy correlation function, turns out not to decrease with the density of source galaxies (and thus with the depth of the observations). In our mathematical framework, the contribution of the small asymmetry in the Jacobian matrix induced by double lensing is retained consistently up to 2 orders in the weak-lensing asymptotic expansion. The analysis is then checked and exemplified by means of simulations.
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